41. toward practical opportunistic routing with intra-session network coding for mesh networks

7/31/2019 41. Toward Practical Opportunistic Routing With Intra-Session Network Coding for Mesh Networks

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Toward Practical Opportunistic Routing with

Intra-session Network Coding for Mesh Networks

Bozidar Radunovic1, Christos Gkantsidis1, Peter Key1, Pablo Rodriguez2

1 Microsoft Research 2 Telefonica ResearchCambridge, UK Barcelona, Spain

{bozidar,chrisgk,peter.key}@microsoft.com [email protected]

AbstractWe consider opportunistic routing in wireless meshnetworks. We exploit the inherent diversity of the broadcastnature of wireless by making use of multi-path routing. Wepresent a novel optimization framework for opportunistic routingbased on network utility maximization (NUM) that enables usto derive optimal flow control, routing, scheduling, and rateadaptation schemes, where we use network coding to ease therouting problem. All previous work on NUM assumed unicasttransmissions; however, the wireless medium is by its nature

broadcast and a transmission will be received by multiple nodes.The structure of our design is fundamentally different; thisis due to the fact that our link rate constraints are definedper broadcast region instead of links in isolation. We proveoptimality and derive a primal-dual algorithm that lays the basisfor a practical protocol. Optimal MAC scheduling is difficultto implement, and we use 802.11-like random scheduling ratherthan optimal in our comparisons. Under random scheduling, ourprotocol becomes fully decentralized (we assume ideal signaling).The use of network coding introduces additional constraints onscheduling, and we propose a novel scheme to avoid starvation.We simulate realistic topologies and show that we can achieve 20-200% throughput improvement compared to single path routing,and several times compared to a recent related opportunisticprotocol (MORE).

Index Termswireless mesh networks, network coding, op-portunistic routing, broadcast, multi-path routing, flow control,fairness, rate adaptation

I. INTRODUCTION

One of the main challenges in building wireless mesh net-

works ([1], [2], [3]) is to guarantee high performance despite

the unpredictable and highly-variable nature of the wireless

channel. In fact the use of wireless channels presents some

unique opportunities that can be exploited to improve the

performance. For example, the broadcast nature of the medium

can be used to provide opportunistic transmissions as sug-

gested in [4]. In addition, in wireless mesh networks, thereare typically multiple paths connecting each source destination

pair, hence using some of these paths in parallel can improve

performance [5], [6].

However most of the existing work on optimal wireless

protocol design (c.f. [7]) ignores the broadcast nature of the

channel. Instead, a transmitter selects a priori the next-hop

for a packet, and if the selected next-hop has not received the

packet, the packet is retransmitted (even though another next-

hop neighbor may have received it correctly). The routing is

not opportunistic (as in [4]) and the diversity of the broadcast

medium is ignored.

The main focus of this paper is the optimal use of both

multiple paths and opportunistic transmission. We use intra-

session network coding [8] to simplify the problem of schedul-

ing packet transmissions across multiple paths, as others have

done to [5], [6], [9]. We propose a network optimization

framework that optimizes the rate of packet transmissions

between source and destination pairs.

In order to use the resources of a wireless mesh networkefficiently, the system needs to take into account: (a) the

existence of multiple paths, (b) the unreliable nature of wire-

less links, (c) the existence of multiple transmission powers

and rates (which in turn affects the probability of correct

packet reception), (d) the broadcast nature of the channel, (e)

competition among many flows, (f) fairness and efficiency.

Observe that optimizing across all these parameters implies

optimizing across multiple layers of the networking stack;

for example, the choice of transmission power and rate is

typically done at the physical layer, whereas coordination

among different flows is typically done at the network layer.

As we shall see, it is important to perform such cross-layer

optimizations to achieve optimal performance.We use an optimization framework to design a distributed

maximization algorithm. We account for transport layer con-

trols and address questions of fairness by maximizing the

aggregate utility of the end-to-end flows, where we associate

a utility function U() with a flow. Because we use networkcoding, our optimization leverages existing theory [10], [9].

Our algorithm is a primal-dual algorithm [11]. The primal

formulation expresses the optimization problem as a function

of the rates of the various flows in the network; the dual for-

mulation uses as variables the queue lengths (per flow and per

node). The main advantage of using the dual formulation of the

optimization problem is that the dual variables (also referred

as shadow prices) relate to queue lengths and can be directlyused by back-pressure algorithms for flow control [12], [7].

As a simple example, a large number of queued packets for

a particular flow at an internal node can be interpreted that

the path going through that node is congested and should

be avoided. The main advantage of using the primal-dual

formulation is that it adapts the primal variables (i.e. flow

rates) more slowly, hence, allows TCP-like window-based

rate control modeling (as originally mentioned by Erylimaz

et al. [12]). We propose a novel algorithm for cross-layer

optimization and prove, using Lyapunov functions, that it

converges to the optimal rate allocation.


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Despite using similar optimization techniques to prior work

(e.g. [12], [7], [13], [14], [11]), the solution to our problem

is very different. We define rate constraint for each set of

broadcast receivers. Consequently, dual variables are related

to these broadcast sets, and allow us to adjust the level of

opportunism as a function of a congestion in the rest of the

network.

The proposed optimization framework is difficult to im-

plement; indeed, the joint scheduling, rate and power control

problem is NP-hard [15]. Additionally, current wireless MAC

protocols use uncontrolled randomized channel scheduling.

We propose a distributed heuristic based on the optimal algo-

rithm. We show that, even in the absence of optimal channel

scheduling, the other aspects of the optimization problem

(i.e. flow selection and transmission rate selection) still give

performance advantages. Hence, our heuristic hints toward an

implementation in practical systems. The fundamental idea

behind our algorithm (and, of its distributed implementation)

is to assign credits to nodes, transfer credits between nodes,

and schedule on the basis of credits (see Sec. III for more

details).The main contributions of our paper are as follows:

We propose a network wide optimization algorithm thatmaximizes rate-based global network performance, and ex-

tends previous work by incorporating broadcast/opportunistic

routing, multi-path routing, and fairness/rate control (Sec-

tions II and III). We introduce a notion of virtual packets,

called credits, that enable us to decouple routing and flow

control from actual packet transmissions and delivery. We

prove the optimality of the algorithm.

Based on the optimization algorithm, we give a dis-tributed implementation (assuming ideal signaling) of routing,

rate adaptation, and flow control for networks with random

scheduling (Section III-C) that outperforms existing algo-

rithms. We prove that our algorithms extends and outperforms

a recent proposal, MORE [5]. The distributed algorithm can

be used with the current 802.11 MAC, and indeed is MAC

independent.

Practical network coding schemes use finite generation sizes.We show that a naive approach for scheduling generations may

lead to starvation. We propose a novel heuristic and we demon-

strate that it circumvents network starvation (Section IV-A).

We demonstrate that rate selection is important for opti-mizing performance in 802.11a networks (Section IV-B). We

confirm the findings from [5] that such optimizations are not

necessary for 802.11b networks.

Using simulation on realistic topologies, we show we canachieve 20-100% throughput improvement with our distributed

implementation compared to single path routing, and 20-300%

compared to MORE [5] (Section V).1

The rest of the paper is organized as follows. Section II de-

scribes the model we are using. Section III gives the optimiza-

tion problem, describes an approximation of the problem that

1Observe that MORE optimizes the number of delivered packets for flowsin isolation, and,when multiple flows are active, may perform worse thansingle path routing w.r.t. rates.

3

1

2

4

p12

p13

p24

p34

C12

C13

y12

y13

C1{2,3}

Fig. 1. A network with 4 nodes is shown on the left. For example, anactivation profile {(1, 2), (3, 4)} depicts a profile where nodes 1 and 3 aretransmitting, node 2 is receiving a packet from node 1, while node 4 isreceiving from node 3. Profile {(1, {2, 3})} depicts node 1 transmittingand nodes 2 and 3 receiving (the same) packet from node 1. The feasiblerate region for (y12, y13) is given on the right, described by inequalities:

jJ y1j C1J for all J {2, 3}.

can be computed in a distributed system, and compares with

a recent proposal for multi-path routing. Section IV discusses

some practical issues, namely the effect of limited generation

sizes, and the effect of randomized 802.11-compatible channel

scheduling. Section V evaluates the performance of our system

using simulation. Section VI provides related work.

I I . MODEL

In this section we introduce our notation, denoting vectors

in bold typeface. We extend the model of a wireless erasure

network developed in [16] to include multiple flows.

A. PHY And MAC Characteristics

We consider a network comprising of a set of nodes N, N =|N |. Whenever a node transmits a packet, several nodes mayreceive it. We model packet transmission from node i to aset of nodes Ki N with a hyperarc (i, Ki). We define an

activation profile S = {Sl} to be a set of hyperarcs active atthe same time. There may be several constraints on feasible

activation profiles. For example, a node may be limited to

receive from but one node, or transmit to only one node at a

time. The only condition we shall impose is that a node can

be the source of only one hyperarc in one activation profile.

All the other constraints can be expressed through reception

probabilities and our model is general enough to incorporate

them (in particular, it is possible that a node transmits while

some information is being sent to it, in which case we shall

set the probability of successful reception to 0; see below for

details). We denote by S the set of feasible activation profilesand let SRC(S) = {i N |Ki N, (i, Ki) S} be the

set of transmitters in activation profile S.Each transmission has two associated parameters, power

P P and rate R R, where P is the set of allowedtransmission powers (e.g. P [0, PM], where PM is given byregulations) and R is sets of available PHY transmission rates,defined by supported spreading, coding, and modulations.

Consider an activation profile S in which node i transmitsto set of nodes Ki, and suppose node i is transmitting withpower Pi and rate Ri. We can associate power vector P =(Pi)iN rate vector R = (Ri)iN to these transmissions. LetTij(P, Ri, S) be an indicator of a random event such thatTij(P, Ri, S) = 1 if a packet is successfully delivered from


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2

5

4

Fig. 2. Forwarding example. Lines connect nodes that can exchange packets.

Opportunistic routing implies multiple-paths. Possible paths from 1 to 5 arefor example (1, 2, 3, 4, 5), (1, 2, 3, 5), (1, 2, 4, 5), (1, 3, 4, 5), (1, 3, 5). Path(1, 2, 1, 3, 5) is also possible but will be typically eliminated by a routingprotocol due to suboptimality.

i to j Ki. It depends on the packet transmission power Piand rate Ri, as well as on the interference from concurrenttransmissions described through P and S. We also denote by

TiJ(P, Ri, S) = 1 jJ

(1 Tij(P, Ri, S))

the indicator of the event that at least one of the nodes from

J, (J Ki = ) receives a packet. Consequently

piJ(P, Ri, S) = Prob(TiJ = 1).

By convention, we assume piJ(P, Ri, S) = 0 if J Ki = ,for (i, Ki) S. Our model does not require any assumptionson channel conditions; in particular, events Tij and Tik arenot assumed independent.

We can now calculate CiJ(P, Ri, S), the average numberof packets per unit time conveyed from node i to any of thenodes in J N. We have

CiJ(P, Ri, S) = RipiJ(P, Ri, S) (1)

B. Traffic, Forwarding And Flow Scheduling

There is a set of unicast end-to-end flows C in the network,and each flow c C has a source and a destination nodeSrc(c), Dst(c) N respectively. We denote by fc the rate offlow c.

Opportunistic routing does not a priori rely on a notion

of a path or a route. Consider the example of Figure 2: a

packet going from node 1 to node 5 may be relayed by any of

the nodes 2, 3, 4, depending on which node happens to hear

it. Thus implicitly, opportunistic routing implies multi-path

routing without pre-specifying the path. Formally, a packet

from node i can reach node j if there exists an activationprofile that allows such a transmission (there exists S S such

that (i, Ki) S and j Ki). The goal of our optimizationframework is to derive how many packets should be forwarded

by each of the nodes.

In practice, an external routing protocol can be combined

with the opportunistic approach to further improve efficiency.

Consider again the example from Figure 2: if a packet is

broadcast by node 2 destined for 5, it may be received and

potentially forwarded by node 1. Our opportunistic routing

protocol will, as described latter, eliminate such events, but

will take some time to discover them. Instead, one may

use a readily available external routing protocol to promptly

eliminate obviously suboptimal paths (e.g. path (1, 2, 1, 3, 5)

from Figure 2). Our model can be easily extended to include

constraints imposed by an external routing protocol, but we

omit such extensions to simplify the exposition. We do con-

strain the set of available paths in the numerical results, as

explained in Section V.

A node that transmits a packet cannot control who will

receive the transmitted packets due to channel randomness.

A node that receives a packet has to decide whether it

will forward it or not. This decision is made through credit

assignment, described in Section II-D.

Whenever a node is about to transmit, it needs to decide

which flow it will transmit. This is defined through a flow-

scheduling profile matrix A. If node i transmits a packet fromflow c we set Aic = 1, otherwise Aic = 0. We say that a flowscheduling profile is valid if for each i N there exists onlyone c C such that Aic = 1. A denotes the set of all validflow scheduling profiles.

To illustrate the use of flow scheduling profile, consider the

example in Figure 1 having two flows C = {1, 2}, both from1 to 4. The number of packets for flow c sent by node 1 and

received by node 2 depends not only on how often (1, 2)is scheduled, but also on how often (1, {2, 3}) is scheduledto transmit a packet from c. This is why the flow schedulingdecision is assigned to a node instead of a link, which is in

sharp contrast to [17], [18], [12].

C. Network Coding

We assume network coding per flow is used [16], [5]. The main

benefit of network coding is that it facilitates scheduling. If

the same packet is received by several nodes, a mechanism

is needed to prevent two or more nodes forwarding the same

packets [4]. To eliminate this problem, each relay forwards a

random linear combination of all previously received packetsfrom the same flow. It has been show in [9] that the ran-

dom linear combinations received at the destination will be

independent with high probability, hence the packets can be

restored.

Ideally, network coding should be performed across the

entire flow. However, this is not practical. Instead, packets

are divided in generations and only packets from the same

generation are combined. For more details see [16], [5]. In

Section III we analyze the optimal network design assuming

very large generation sizes (as in [16]); we address finite

generation sizes in Section IV.

D. Credits

As remarked on earlier, whenever a packet is transmitted,

it may be received by several nodes, and it is important

to decide which should forward packets, to avoid redundant

transmissions (as explained in [19], [5]).

We introduce the concept of credits, which is similar to the

control decision variable of Neely [19]. One credit is created

for each packet at the source node. Credits are identified with

a generation, not a specific packet. They are conserved until

they arrive at the flows destination. In this way we guarantee

that the destination will receive as many linear combinations of


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the packets as the number of packets generated at the source,

and hence will be able to decode the packets.

Credits are interpreted as the number of packets of a specific

flow to be forwarded by a node. By controlling the rate of

credits we control the rate of packets forwarded by next-hop

relays. Consider again the example from Figure 1. Node 1

should adapt the rate of packets forwarded through 2 and 3

not only as a function of link qualities p12 and p13, but alsoas a function of p24 and p34, the quality of paths from 2 and3 to the destination 4. For example, if p24 p34 then node 2should not forward any packet, regardless of how many it has

received. Node 1 cannot control what nodes 2 and 3 receive

due to randomness of the channel. Instead, node 1 sends a

credit to node 2 (or node 3) whenever it wants node 2 (or

node 3) to forward a packet.

The main advantage of the credit scheme is that it simplifies

scheduling. Credits are declarations of intent. The actual

packet transmissions may occur at arbitrary time instants. Due

to the use of network coding, we only need to ensure that the

total number of packets per generation transmitted between

each two nodes corresponds to the number of credits. Thus,scheduling is done at a generation level and not at the packet

level, incuring significantly smaller overhead (especially when

the generation size is large).

In practice, credits can be piggybacked with packet trans-

missions. The receiving node only updates its credits when a

successful packet transmission actually occurs. In this work

we assume there is an ideal (no loss and no delay) signaling

plane that transmits credits and feedbacks.

As each credit delegates one packet to a node, we may

express all the rates in the system in terms of credits. For

example, ycij is the rate of credits of flow c passed fromnode i to node j, and it equals the rate of innovative linear

combinations of packets of flow c delivered from i to j.Theorem 1 shows that the rate of independent packets receivedat a destination of each flow will correspond to the number of

credits delivered, when the generation size is large.

E. Dynamics And Stability

We further assume the system is slotted in time. In each slot

t = 0, 1, . . . a medium access protocol assigns an activationprofile S(t) and a flow-scheduling profile A(t), and to eachtransmitter i SRC(S(t)) we assign transmit power Pi(t)and rate Ri(t). Let y

cij(t) be the number of credits for flow c

transmitted from node i to node j during slot t, and let xciJ(t)denote the number of packets of flow c actually delivered from

i to any of the nodes in J during slot t (as if all nodes in Jare grouped as a single receiver). Let fc(t) be the number offresh packets/credits generated at the source of flow c.

Note that, because each successful packet delivery is always

associated with a credit transmission, we look at credit queues.

Let qci (t) be the amount of credits of flow c queued at nodei. The system is stable if every queue size is bounded. Wedefine stability more formally in Section III-D.

F. Constraints Of The Model And Possible Generalizations

We make several simplifying assumptions to make the

analysis tractable. Firstly, we assume that the system is slotted.

All operations within a slot occur concurrently and instantly.

Secondly, we assume the perfect signaling. There is no loss

or delay in signaling messages (credit transfers and acknowl-

edgments).

Our results can be extended to consider imperfect signaling

and arbitrary but limited delays in the system (see for example

[20, Part 2] on a discussion how does a delayed feedback affect

the speed of convergence). Also, see [21] for a practical imple-

mentation of a back-pressure based system and its interaction

with TCP.

III. OPTIMAL FLOW CONTROL FOR FAIRNESS

In this Section we introduce the optimization problem

(Sec. III-B), propose an algorithm for solving it (Sec. III-C),

and prove that the algorithm converges (Sec. III-D). Sec.III-A

introduces some further notation that is needed for the descrip-

tion of the optimization problem. Finally, in Sec. III-E we

compare our algorithm with the MORE algorithm proposed

in [5].

A. Feasible Average Rate SetIn this section we define a set of constraints on average rates

in the system. Assume an assignment of average end-to-end

rates fc, for each flow c, and denote the average rate vector byf = (fc)cC. The vector of rates is valid under the followingthree conditions. First, traffic at node i is stable if the totalingress traffic is smaller than the total egress traffic, which we

write as j=i

ycji + fc1{i=Src(c)} j=i

ycij , (2)

for all i = Dst(c), where 1x = 1 if x is true, 0 otherwise, andwhere yij 0.

Second, traffic at each broadcast region is stable if we donot receive more credits than we can actually forward (see

also Fig. 1): jJ

ycij xciJ, for all J N. (3)

Recall that ycij is the average number of credits of flow c nodei assigns to node j and xciJ is the average number of packetsof flow c actually delivered from i to any of the nodes in J.

Finally, we define scheduling constraints. A schedule is a se-

quence {(S(t),R(t),P(t),A(t))}t0 which defines schedul-ing profile S(t), routing profile A(t) and power and rateallocations R(t),P(t) in each slot t 0. Since we are

interested in long-term average rates, we define

S,R,P,A = limT

1

T

tT

1{S(t)=S,R(t)=R,P(t)=P,A(t)=A}

to be the fraction of time the network uses scheduling profile

S, routing profile A and power and rate allocations R,P.By definition, S,R,P,A 0 and

S,R,P,A S,R,P,A 1.

The schedule defined by {S,R,P,A}S,R,P,A is stable if itcan support broadcast traffic {xciJ}i,J,c, and we write thescheduling constraints as

xciJ

S,A,R,P

S,R,P,AAicCiJ(P, Ri, S) (4)


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We use the following characterization of feasible rates from

[9]:

Definition 1: Vector f is said to be feasible if each flow ccan transport information from Src(c) to Dst(c) at averagerate fc.

Theorem 1: Let F be the set of average end-to-endrate vector f = (fc)cC such that there exists vectorsy = (ycij)i,jN,cC , x = (x

c

iJ

)iN,JN,cC, and =(S,R,P,A)SS,RRN,PPN,AA that satisfy (2), (3), and (4)subject to S,R,P,A 0 and

S,R,P,A S,R,P,A 1. The

vector f is feasible when coding generation size goes to infinity

if and only if it belongs to F. Moreover, the set of feasibleend-to-end rates F is convex.

Proof: Proof of feasibility follows directly from [9]. The

set is convex since all constraints are convex.

B. Utility Maximization

For each flow c C we define a utility function Uc() tobe a strictly concave, increasing function of end-to-end flow

rate fc. The utility of flow c is then Uc(fc). For example,

Uc(fc) = log(fc) represents proportional fairness [22] andUc(fc) 1/fc approximates TCPs utility [11]. The goalof utility maximization is to achieve a trade-off between

efficiency and fairness. Proportional fairness is an example

of such an approach [22].

We can write the network-wide optimization problem as

max

cC Uc(fc)

s.t. f F.(5)

Since set F is convex and the objective is strictly concave,there exists a unique solution f to the maximization problem.

Corresponding y,x also exist but are not necessarily unique.Let us denote with ci and

c

iJ

the Lagrangian multipliers as-

sociated with inequalities (2) and (3), respectively. To simplify

the notation we will also define cDst(c) = 0. We can write theKKT conditions at the optimal point for the constraints (2)

and (3)

ci

j=i

ycij j=i

ycji fc 1i=Src(c)

= 0, (6)

ciJ

xciJ

jJ

ycij

= 0, (7)

Also, combining the condition that the rates are non-negative

(fc 0) and the gradient KKT condition, we can writefc

Uc(f

c )

cSrc(c)

= 0. (8)

Note that here we do not write the KKT for the constraint (4)

but we keep it in the explicit form (see for example (10)).

We see that ci = 0 if egress traffic is larger than ingresstraffic and there is no accumulation of credits at node i. Henceintuitively we can relate ci to q

ci (t), the number of credits for

flow c queued at i. Similarly we can relate ciJ to the numberof packets queued for broadcasting at i. In Section III-C weexpress this relationship more formally. We will also use (8)

to develop a flow control algorithm.

As a consequence of KKT, and by optimizing the dual

problem [23] one can derive the conditions

0 ci cj

JN | jJ

ciJ, (9)

CiJ =CiJ(P, Ri , S

), (10)

(P

, R

i , S

) = argmax{(P,Ri,S)}i max

cJ

c

iJCiJ(P, Ri, S).

We will use (10) in Section III-C to derive the optimal

scheduling.

C. Maximization Algorithm

We now present an algorithm that converges to the optimal

value of (5).

Node and Transport Credits : Recall that qci (t) is the amountof commodity c credits queued at node i. We call such creditsnode credits. In addition, let wciJ(t) be the number of creditsof commodity c queued at i and corresponding to packets that

have to be delivered to any of the nodes in J (as previouslydecided by the credit transmission scheme). We call these

transport credits.

When a credit for flow c is transferred from node i to nodej, we decrease qci , we increase q

cj , and we increase w

ciJ for

all J j (all by one unit). Note that this transfer happensinstantly, and before a corresponding packet has actually been

transmitted. We decrease wciJ when a packet from flow c isactually delivered from i to any of the nodes in J. The amountof node credits is conserved: when a responsibility for a credit

is transfered from i to j we decrease qci and increase qcj .

Transport credits are of a different nature. They are created

when a forwarding decision is made and are destroyed when

the actual delivery takes place. Transport credits are local and

they are never transferred between nodes.

Routing protocol: Node credits represent intentions of packet

transmissions and a routing protocol describes when and how

such node credits transferred. Let ycij(t) be the number of nodecredits for flow c transferred from node i to node j at timet and let us define wcij(t) =

XN | jX w

ciX(t). A back-

pressure between nodes i and j is defined as

zcij(t) = qci (t) w

cij(t) q

cj(t), (11)

the difference between the excess credits queued of flow c atnode i not destined for node j (qci w

cij) and the node credits at

node j (qcj ). Back-pressure in (11) includes the credits queuedat the next hop as well as the credits assigned to different

broadcast regions, which is the main difference compared to

previous work [17], [18], [12].

A credit is transferred from i to j with the followingdynamics

ycij(t) =qci (t)

|{k | zcik(t) > 0}|1{zcij(t)>0}, (12)

where 1{x>0} is 1 if x > 0 or 0 otherwise, and |{k | zcik(t) >

0}| is the number of neighbors of i that have a positive back-pressure for flow c. The queue qci (t) then has the following


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dynamics in time

qci (t + 1) = qci (t) + fc(t)1i=Src(c) +

j

ycji(t) j

ycij(t).

Note that by definition

j ycij(t) q

ci (t) hence q

ci (t) is

always positive.

The intuition is as follows. We transfer a credit from i to j

only if the back-pressure is positive. Moreover, we share allof the available credits qci (t) equally among all neighbors witha positive back-pressure.

Scheduling, rate and power control: The optimal centralized

scheduling, rate and power control algorithm is the tuple

(S(t), P(t), R(t), A(t)) that solves the following optimizationproblem

W Ci(t,P, Ri, S) = maxc

J

wciJ(t)CiJ(P, Ri, S), (13)

(S(t),P(t),R(t)) = argmaxS,P,R

iN

W Ci(t,P, Ri, S), (14)

CiJ(t) = CiJ(P(t), Ri(t), S(t)), (15)ci (t) = argmax

c

K

wciK(t)CiK(t), (16)

Aic(t) = 1{c=ci (t)}, (17)

xciJ(t) = AicCiJ(t). (18)

Intuitively, (13) follows from (10) and the fact we can equate

ciJ and wciJ, since the transport credit update equation (21)

corresponds to the gradient update equation of ciJ. The otherupdate rules follow directly from KKT conditions.

Equations (13)-(18) represent a joint scheduling, rate, and

power control problem. We find the optimal scheduling, power

and rate control (S(t), P(t), R(t)) by solving (14). Then,equation (16) is used to select which flow will be transmittedby each node in slot t.

The main novelty in our approach is that we explicitly

incorporate all broadcast regions in the scheduling algorithm in

Equation (13) through broadcast transport credits wciJ(t). Thisis in contrast to previous works on back-pressure [17], [18],

[12], [14] that are not able to exploit the broadcast diversity. It

is only [19] that consider network optimization with broadcast

diversity, but using different optimization techniques. Also,

unlike [5], [4], we are able to make a trade-off between

broadcast diversity and network congestion.

Another difference with respect to [17], [18], [12], [14] is

that, as explained in Section II-B, we cannot decouple theflow selection process A(t) and routing/scheduling/rate/powercontrol. Also, unlike in [17], [18], [12], [19], we do not

explicitly use back-pressure information for scheduling in (13)

- (18); instead we use transport credits wciJ(t).

The algorithm (13)-(18) is centralized. Observe that all

equations except (14) and (15) use local information only.

Hence, with the exception of (14) and (15), the problem could

have been solved with a distributed algorithm. We use this

observation to propose a heuristic based on modified rules

(14) and (15), and to derive a practical, distributed protocol

that is presented in Section IV.

Flow control: The optimal flow rate at the source, fc(t) canbe calculated using a primal-dual approach, as in [12]

fc(t + 1) =

fc(t) +

Uc(fc(t)) qcSrc(c)(t)

+, (19)

where [x]+ = max{x, 0}. Each flow adapts its rate basedon the previous rate and current number of credits queuing

for transmission at the source node for that flow (qSrc(c)

).

The primal-dual approach well describes additive-increase

multiplicative-decrease transport protocols, like TCP [11].

D. Convergence Of The Algorithm

We now consider a fluid model of the system, and show that

it converges to the optimal point. Analysis of a discrete-time

model can be derived from our fluid-model analysis, using a

similar approach to [12].

We assume that time is continuous and that queue evolutions

are governed by the following differential equations

qci (t) =fc(t)1i=Src(c) +

j

ycji(t) j

ycij(t)qci (t)0

(20)

wciJ(t) =

jJ

ycij(t) xciJ(t)

wciJ(t)0

(21)

where (x)yz = x if y z and (x)yz = 0 otherwise.Similarly, flow rate evolution in the fluid-model is given by

fc(t) = Uc(fc(t)) q

cSrc(c)fc(t)0

. (22)

We next prove that the algorithm presented in Section III-C

stabilizes the system with flow rates that maximize the opti-

mization problem (5).

Definition 2: We say that link (i, j) is active for flow c ifthere exist a finite number T such that for each t that satisfiesycij(t) > 0, there exists t

, t < t < t+T such that ycij(t) > 0.

Theorem 2: Starting from any vectors f(0),q(0),w(0) andapplying rules (11)-(22), the rate vector f(t) converges to f

as t goes to infinity. Furthermore, queue sizes qci (t), qcj(t) and

wcij(t) on all active links (i, j) for flow c are bounded, andconverge to the shadow prices ci ,

cj and

ciJ respectively.

Also, if a node is completely disconnected from the rest of

the network, or in any way not used by a flow, credits willneither arrive to nor will leave from the node. Thus technically,

we cannot guarantee that an arbitrary initial number of credits

at this node will converge to any particular value. Instead, we

consider only links that have at least some traffic through-

out the network run-time (called active links, formalized in

Definition 2).

The proof uses a Lyapunov function with stability defined

on the set of active link, and we show that on all active links

that carry a positive amount of traffic, the delays are bounded

and hence the system is stable. The details are given in the

Appendix.


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7

E. Comparison with MORE

In this section we compare the performance of our algorithm

with the MORE forwarding algorithm described in [5], [24].

We summarize it here for sake of completeness. Consider a

single flow and the delivery of a single packet from the source

of the flow Src to the destination of the flow Dst. Let i bethe number of transmissions made by node i to successfully

deliver the packet. The goal of MORE is to minimize thenumber of transmissions [24, Eq.(1)-(4)]:

argminiN

i (23)

s.t.jN

yij jN

yji = 1{i=Src} (24)

yij 0, (25)

iCiJ jJ

yij. (26)

Note that the MORE forwarding algorithm is designed for a

single flow. As the authors note in [5], [24], its performance

drops as the number of flows increases.Our algorithm converges to the optimal solution of the

optimization problem (5) (as shown in Theorem 2), hence

MORE is at best as good as our algorithm. We first show

under what conditions MORE is guaranteed to give the optimal

solution. We then illustrate by two examples that MORE can

yield strictly suboptimal rate allocations.

Theorem 3: If there is only one flow in the system, if

transmission rates and powers of all nodes are fixed and if only

one node can transmit at a time (that is |SRC(S)| = 1 for allS S), MORE and our algorithm give the same performance.

Proof: Since only one node can transmit at a time, we

have S = N. Furthermore, transmission powers and rates are

fixed, hence (3) and (4) reads as

jJ yij iCiJ. We alsoomit c as there is only one flow in the system.

We start with the MORE optimization problem (23) - (26)

and we introduce f = 1/

iN i

, yij = fyij and i =f i. The optimization (23) - (26) is then equivalent to

min1/f (27)

s.t.jN

yij jN

yji = f1{i=Src} (28)

iCiJ jJ

yij , (29)

iN

i = 1, (30)

which is exactly the optimization problem (5).

We next give two examples where the performance of

MORE is strictly suboptimal. Consider a hexagonal network

depicted in Figure 3. Let us first consider a case with a single

flow f1, (f2 = 0), and where p12 = p24 = p46 = 0.8and p13 = p35 = p56 = 0.2. Since not all links interfere,the conditions of Theorem 3 are clearly not satisfied. The

optimal rate allocation that maximizes (5) is f = 0.313 withy1246 = 0.251 and y1356 = 0.063. However, MOREwill transmit all packets over the path 1 2 4 6, hencethe total rate will be fMORE = 0.267, some 15% less than the

optimal. Intuitively, the reason why MORE is suboptimal is

that it does not consider possibility that links 3 5 and 5 6transmit in parallel with 4 6 and 2 4. (Recall that MOREsgoal is to minimize the number of transmissions and not to

maximize the flow rate.) It will then conclude that forwarding

any packet to 3 is largely suboptimal, since links 3 5 and5 6 are of a bad quality. Thus, ifp35 and p56 are sufficientlysmaller than p12, p24 and p46, as in this example, MORE willnot use route 1 3 5 6 at all.

In the second example we again consider the same hexag-

onal network but with two flows (f1, f2) active. Since MOREdoes not take into consideration contention among flows, it

will again assign all traffic to the path 1246. This trafficwill contend with the traffic from flow 2, and feasible end-to-

end rate allocations have to satisfy 3f1+ f2 = p. Note that therouting scheme is fixed by MORE and does not depend on flow

control applied by transport layer. If for example the transport

layer on top of MORE is designed to maximize log utility,

the optimal rates will be fMORE1 = p12/6 = 0.133, fMORE2 =

p12/2 = 0.4. On the contrary, our distributed algorithm will

adapt routing to contentions among flows, and it will assigny1 = 0.12, y2 = 0.03, f1 = 0.15 and f2 = y3 = 0.4. As wecan see, our algorithm balanced flow 1 by decreasing y1 andincreasing y2. As a result, the rate of flow f2 stayed the samewhile the rate of f1 increased.

IV. PRACTICAL ISSUES

In this section we consider two practical issues that concern

implementation of the protocol proposed in Section III in a

mesh network: finite coding generation size and rate adaptation

for randomized scheduling. We leave other practical issues,

such as the effect of delayed feedback, for future work.

A. Finite Generation Size

Previous results assume that generation size used for network

coding tends to infinity (see Theorem 1). Practical reasons,

such as the complexity and performance of decoding, and

header overhead for storing the coefficient vector, require us

to limit the size of the header; some practical systems limit

the size to 32 bytes [5], [6]. We now modify our optimization

framework for a finite generation size.

3

1

2

6

5

4

p12

p13

p35

p24

p46

p56

Wall

y2

y1

y3

Fig. 3. A network with 6 nodes. Due to a dividing wall, nodes 2 and4 do not interfere with nodes 3 and 5. The set of activation profiles isS = {{(1, {2, 3})}, {(2, 4)}, {(3, 5)}, {(4, 6)}, {(5, 6)}, {(2, 4), (3, 5)},{(2, 4), (5, 6)}, {(3, 5), (4, 6)}}. There are two flows in the system, flowf1 = y1 + y2 which is assigned 2 routes (y1 = y1246 andy2 = y1356) and flow f2 = y3 which is assigned a single route(y3 = y24).


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8

3

1

2

p12

p13

p23

w12(0, 1) = 0,

w12(0, 2) = 5,

w1{2,3}(0, 1) = 0,

w1{2,3}(0, 2) = 4,

w13(0, 1) = 1,

w13(0, 2) = 0

Fig. 4. A simple example of a network with nodes {1, 2, 3} and a singleflow, going from 1 to 3 (directly or via node 2), where set p12 = p23 = 1and p13 is close to 0. The values of corresponding transport credits wiJ(t, g)are given on the right at time t = 0 for generations g = 1, 2 (in the examplewe assume only two generations are in the networks).

Let G be the set of generations. Let us define qci (t, g) andwciJ(t, g) be the number of node credits and transport creditsfor generation g G of flow c queued at i. Similarly, we definefc(t, g), ycij(t, g), x

ciJ(t, g) as before but with respect to gen-

eration g (thus we have for example ycij(t) =

gG ycij(t, g),

and by analogy for the other variables). The encoding pro-

cesses fc(t, g) are defined at the source. For example, if thesize of each generation is G, we havet

fc(t, g)dt = G forall g G.

Extending the distributed maximization algorithm from Sec-

tion III-C to this setting is not straightforward. For example, a

naive way to modify scheduling rule (18) would be to schedule

the oldest generation available

gi (c, t) = min{g | (J)wciJ(t, g) > 0}, (31)

that is xciJ(t, g) = CiJ(t), if c = ci (t) and g = g

i (c

i (t), t).

However, this rule may yield poor rates.

To see why, consider the example of Figure 4 where one

link (1 3) has very poor quality compared to the others.

For simplicity we assume there are only two generations inthe system and we assume that node credits qci (t) are constanthence only the transport credits are the focus of the scheduling

policy. The oldest generation that has a transport credit at node

1 is generation 1. Thus, when node 1 is selected to transmit, it

will transmit a packet from generation g1(t) = 1, according to(31). On one hand, node 2 will certainly receive the transmitted

packet but, since w12(0, 1) = w1{2,3}(0, 1) = 0, node 2 doesnot need more packets from generation 1 and the receivedpacket will be useless. On the other hand, node 3 is not likely

to receive the transmitted packet as p13 0. Therefore, inthe next slot w13(1, 1) will most probably remain at 1 andgeneration one is again selected for transmission. The same

situation will repeat until the transmitted packet is finallyreceived by node 3, and only then we will be able to transmit

a packet from generation 2.

It is easy to see that in such a scenario we may completely

starve the network. Instead of benefiting from diversity, the

opportunistic routing acts as a hindrance. The main reason

for starvation is the fact that the naive approach implicitly

restricts scheduling diversity over a single generation. Ob-

serve that the problem persists even when the number of

generations increases and the queues build up; the naive

approach tries to maintain the optimal traffic splitting strictly

per each generation instead on a cross-generation, long-term

average. Nevertheless, there is no reason why we should not

be able to exploit the diversity of link 1-3, because even a

few occasional packets transmitted over that link will improve

overall performance.

Finding a jointly optimal coding and scheduling strategy

that maximizes system utility for finite generation sizes is a

difficult problem. Apart from the scheduling issue, when the

generation size becomes finite, the coding results from [16]

no longer hold. This implies that packets received at the des-

tination may not be linearly independent and Theorem 1 does

not hold. Instead, we propose a heuristic inspired by the proof

of Theorem 2, which minimizes the drift W(f(t),q(t),w(t)).It consists of modifying rules (12) and (18) to

zcij(t, g) = qci (t, g) w

cij(t, g) q

cj(t, g), (32)

ycij(t, g) =qci (t, g)

|{k | zcik(t, g) > 0}|1{zcij(t,g)>0}, (33)

gi (c, t) = argmaxg

J

wciJ(t, g)xciJ(t) 1{wciJ(t,g)>0}, (34)

xciJ

(t, g) = CiJ(t) if c = ci (t), g = gi (c, t)0, otherwise

. (35)

where 1{wciJ(t,g)>0} is true when there are queued transportcredits wciJ for generation g.

Intuitively, the idea behind the heuristic is to transmit the

generation that has the highest chance of being useful. We

select flow c to transmit according to (16). The benefit of sucha transmission is

Kw

ciK(t)CiK(t), the expected decrease in

transport credits. However, this is only true if the generation

size G is infinite. The transmitted packet from generationg will not be useful if the generation size is finite, andsome of the nodes no longer need any more packets from

generation g (that is wciJ(t, g) = 0 for some J, as illustrated

in the example). To maximizes the actual expected decreaseof the amount of transport credits we select a generation gthat maximizes

Jw

ciJ(t, g)x

ciJ(t, g) 1{wciJ(t,g)>0}, which is

indeed (34).

To see how the new policy works, consider again the

example of Figure 4. Unlike the naive policy (31) that se-

lects generation 1 as the oldest generation among all queued

generations, the new policy (34) will select the most useful

generation which is generation 2, which circumvents network

starvation. A detailed explanation of how this policy is derived

is given in the Appendix. Performance simulations for finite

generation sizes are given in Section V.

The policy (33) - (35) needs a slight caveat, in that some

credits from old generations may get stuck in the network(as the credit from generation 1 did in the previous example).

A straightforward extension is to reassign the credits from

the selected generation gi (c, t) and from the oldest queuedgeneration such that the total number of credits is not altered,

and to guarantee in-order delivery. However in practice, unless

a network is very asymmetric, this is not needed, as the

simulations in Section V verify.

B. Rate Adaptation For 802.11-compatible Scheduling

Finding the optimal scheduling rule (14) is an NP-hard cen-

tralized optimization problem, as Sharma et al. show [15].


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9

Some recent research [25], [15], [26] explores decentralized

implementations of similar problems. Applying these ideas to

our setting is outside the scope of this paper, and left for

future work. Instead, we consider a more realistic, suboptimal

scheduling process and we show how our algorithm can be

applied as a distributed heuristic.

We assume that nodes always transmit packet at the full

power Pi(t) = {0, PMAX}, which reflects current practice inmost existing wireless mesh networks deployments. We call a

set of feasible activation profiles S 802.11-compatible if for allS S and for all (i1, J1) S there is no (i2, J2) S suchthat reception probabilities pi1,i2 > 0, (j J2)pi1,j > 0or (j J1)pi2,j > 0. Intuitively, this corresponds to802.11-like protocol with RTS/CTS mechanism. When node i1establishes communication with nodes J1, all nodes involvedin communication send an RTS/CTS. All nodes that hear the

RTS/CTS (p > 0) will be prevented from transmission orreception during the same slot.

Furthermore, we will assume that the underlying scheduling

process {S(t)}t is outside our control, and that it is inde-

pendent of the actions of our protocol. At every time t, thescheduling process will select a set of non-interfering nodesI(t) N to transmit (i.e. for each i, j I(t), pij = 0).Each node i I(t) has a set of possible destinationsJi(t) = {j N |pij > 0, (k I(t), k = i), pkj = 0},which in turns define a schedule S(t) = {i, Ji}iI(t). A setof activation profiles S = {{i, Ji}iI | I P(N)} is clearly802.11-compatible.

With such a schedule S(t), the optimization (13) - (18)simplifies to

(ci (t), Ri (t)) = argmax

c,Ri

KJi

wciK(t)CiK(Ri). (36)

where CiK(Ri) = ES[CiK(Ri, S)] are the average ratesobserved over a long period of time. This optimization can be

easily solved in a distributed manner, locally and separately at

each node. Node i can estimate CiK(Ri) either by probing,or using statistics from previous transmissions. In practice, as

reported in [27], successful transmissions Tij and Tik are oftenindependent for j = k which further simplifies the estimation.The rest of (36) clearly relies only on local information.

Alternatively, if one deploys a form of interference-aware

scheduling (e.g. [28]) which disposes of S(t) in any slot,algorithm (36) can be improved by estimating CiK(Ri, S) foreach scheduling policy S = S(t) separately, instead of usingthe average estimate CiK(Ri). In the simulation part of our

paper we implement the simpler, interference-oblivious form,as given in (36).

Note that for an arbitrary scheduling process {S(t)}t, thedistributed routing (12) and rate adaptation (36) algorithm

do not necessarily minimize the optimization problem (5).

The optimal algorithm will depend on the characteristics of

{S(t)}t, {S(t)}t will depend on our routing algorithm, andit is difficult to characterize these dependences. We present

(12) and (36) as a heuristic that can be used as a prac-

tical implementations of opportunistic multi-path routing in

networks with 802.11-compatible scheduling. We illustrate

by simulations in Section V that in the case of random,

802.11-compatible scheduling, the heuristic (36) outperforms

a conventional, single-path routing approach.

V. SIMULATION RESULTS

We now present simulation results which quantify the per-

formance advantages of the opportunistic routing, scheduling

and flow control algorithms defined in the previous sections.

We are primarily interested in algorithms that can be ap-

plied in 802.11-like mesh networks, where the scheduling

algorithm is not under our control. Hence in our simulations

we used an 802.11-compatible schedule {S(t)}t, as definedin Section IV-B, assuming I(t) is randomly selected amongbacklogged nodes.

We use the roofnet network topology based on 802.11b

cards, given in [4], for our simulations. We further assume

the (802.11-compatible) node-exclusive model with random

channel parameters from [1]). We developed a slotted discrete-

event simulator that implements the routing, flow and rate

control algorithms. A scheduler randomly selects a set of non-

exclusive nodes for transmission in each slot. The amount ofdata transmitted in each slot is proportional to the transmission

rate and the packet loss probability is obtained from [1]

(assuming that a concurrent transmission is allowed; otherwise

it is set to 0). Unless stated otherwise, we assume finite

generation size of 32 and use rules (33) - (35) to select

which generation to transmit. In addition, we allow credit and

packet transmissions by a node only if a node has received

an innovative packet for a given generation since the previous

transmission. We used Uc() = log(), hence the rate allocationthat maximizes (5) is the proportionally fair rate allocation

[22].

We looked at three performance metrics. The first one

is the improvement in total utility

c U(fc)

c U(fc).Allocation f is better than f if the sum is positive. The

proportional fair rate maximizes the optimization problem (5)

hence has the highest utility.2 The second metric is the total

rate improvement

c fc/

c fc. Allocation f is better than

f if the quotient is larger than 1. The proportionally fair

allocation does not always have highest total rate. The third

metric is the Jains fairness index improvement. Jains fairness

index is defined as F I(f) = (

c fc)2/(|C|

c f

2c ) and the

Jains fairness index improvement is F I(f)/F I(f). Note thatthe fairness index can be deceiving as a metric in some cases:

iff has all rates larger than f it may still have smaller fairness

index although the system has clearly improved.

We compared our algorithm with a conventional, single path

routing algorithm, and with the MORE algorithm [5]. To make

the comparison fair, we assumed that the single-path routing

algorithm used the same kind of jointly-optimal routing and

flow-control approach as our scheme, which boils down to

[12], constrained to the best path. In contrast, MORE does

not integrate flow control or flow scheduling with the routing

algorithm. When simulating the MORE algorithm, defined in

[5], [24], we assumed that each source had a large backlog

2Since in the simulations we use random and not the optimal scheduling,the resulting rate allocation does not necessarily have the highest utility.


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10

(a)

0 10 20 30 402

0

2

4

6

Run

Absoluteimprovementinutility

4 flows

8 flows

(b)

0 20 40 60 800.5

1

1.5

2

2.5

Node pairs

Relative

improvementintotalrate

4 flows

8 flows

(c)

0 20 40 60 80

0.8

1

1.2

1.4

1.6

RunRelativeim

provementinfairnessinde

4 flows

8 flows

Fig. 5. Cumulative performance improvement of opportunistic over single-path (fsc - rates using single path, fmc - rates using multiple paths): (a) Absolute

improvement in utility (

c log(fmc )

c log(f

sc )); (b) Relative improvement in total rate (

c f

mc /

c fsc ); (c) Relative improvement in fairness index

(FI(fm)/FI(fs)). We perform the experiments with 4 and 8 concurrent flows. In all cases we ran 100 experiments and sorted them by performanceimprovement. In many cases, for single-path routing, some flow had zero rates for the duration of the simulation, caused by slow convergence; we omittedsuch plots (as they would give an infinite utility difference).

(a)

0 20 40 60 80

0

2

4

6

8

Run

Absoluteimprovement

inutility

4 flows

8 flows

(b)

0 20 40 60 800

1

2

3

4

5

6

Run

Relativeimprovementin

totalrate

4 flows

8 flows

(c)

0 20 40 60 800.5

1

1.5

2

2.5

3

3.5

RunRelativeimprovementinfairnessinde

4 flows

8 flows

Fig. 6. Cumulative performance improvement of our algorithm over MORE (fsc - rates using single path, fmc - rates using multiple paths): (a) Relative

improvement in utility (

c log(fmc )

c log(f

sc )); (b) Improvement in total rate (

c f

mc /

c fsc ); (c) Improvement in fairness index (FI(f

m)/FI(fs)).We perform the experiments with 4 and 8 concurrent flows. In all cases we ran 100 experiments then sorted them by performance improvement.

of packets to transmit, and that each relay performed FIFO

scheduling among packets from different flows.

We ran simulations to obtain end-to-end rate allocations.

Figure 7, (a) illustrates the optimal rate allocations, obtained

by our algorithm, for 8 randomly selected source-destination

pairs on one example. In this case, we can see both utility and

total rate increase if we use the opportunistic routing instead

of the single-path routing, where we benefit from broadcast

and multi-path diversity.

We then ran the previous experiment with 100 random

realizations of 4 or 8 coexisting unicast sessions and compared

the performances of the different algorithms with respect to

the two performance metrics.Single Path vs. Opportunistic: We start by illustrating the

benefits of the opportunistic routing over the single-path rout-

ing in Figure 5. We first look at the network utility. The rate

allocation obtained by the optimal algorithm (Section III-C)

always maximizes the utility. However, this is not the case for

the distributed heuristic (Section IV-B). In our simulations we

saw that in about 90% of the runs, the distributed heuristic for

opportunistic routing achieves higher utility than does single-

path routing. In only about 10% 15% of cases is the utility

for single-path routing higher.

Also, in more than 80% of runs, our decentralized heuristic

achieved higher total rate than the conventional, single-path

algorithm. In more than half of the runs, the total rate has

increased by 20%, and in some cases by over 100%. From

these results we see that there is a significant advantage in

using our opportunistic routing algorithm over the single-path

one. Fairness index also improves in more than half of the

cases.

Decentralized Heuristic vs. MORE: We next compare our

decentralized heuristics with MORE. The results are depicted

in Figure 6. Network utility is increased in about 90% of

the runs. Total rate is increased in almost all of the runs,

sometimes up to a factor of 4 5. The fairness index has also

increased in most of the cases. The performance of MOREdrops with the number of flows.

From these results we can see that in many cases MORE be-

haves worse than the single-path routing. This resonates with

the findings of [5] where the benefits of opportunistic routing

decrease as the number of flows increase (for an explanation,

see Section III-E). MORE is essentially a routing protocol,

whereas our single-path routing algorithm also includes more

intelligent flow control and flow scheduling.

Effects of Finite Generation Size: Figure 7, (b) illustrates

the impact of a finite generation size. The performance drop

is due to imperfect generation scheduling (34) and occasional


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11

(a)

Singlepath Multipath MORE0

1

2

3

4

5

R

ates[Mbps]

(b)

0 20 40 60 800.75

0.8

0.85

0.9

0.95

1

Run

Relative

efficiencyfortotalrate

(c)

0

0.2

0.4

0.6

0.8

1

Nodes

Fractionoftime

54.00Mbps

18.00Mbps

12.00Mbps

9.00Mbps

6.00Mbps

Fig. 7. (a) End-to-end rate allocation example: eight flows among randomly selected source-destination pairs. The vertical axis shows flows rates. Smallbars denote rates per flow. Large bars show total rate. The marks connected by a line gives utilities with an arbitrary scaling (to fit the figure). (b) Relative

efficiency for total rate (

c ffinitec /

c f

infinitec ) for opportunistic routing with a finite generation size of G = 32 . (c) Optimal distribution of PHY rates

for roofnet network with 802.11a cards, for one random selection of 8 source-destination pairs.

linear dependency of received packets. We can see that a finite

generation size of 32 packets can cause a performance drop of

up to 22% with respect to the infinite generation size. 3 How-

ever, a larger generation size would impose more overhead in

transmitting larger coefficients in the packet header.

The Optimal PHY Rate Selection: Finally, we consider

the optimal PHY rate selection at different nodes. We analyze

how frequently each node uses each PHY rate. In the case of

roofnet topology with 802.11b cards we find that in almost

all cases it is optimal to use the highest rate of 11 Mbps,

which confirms the findings from [5]. We then used the SNR

data from roofnet topology and measurements from [29] to

analyze the approximatively optimal performance in the same

network topology with 802.11a cards. The results are depicted

in Figure 7, (c). As expected, the optimal PHY rate selection

is no longer uniform, a consequence of the large number ofavailable rates. This demonstrates the need for an intelligent

PHY rate selection algorithm.

VI . RELATED WOR K

One of the first uses of opportunistic routing for unicast

sessions in wireless mesh networks is presented in [4]. It has

been extended in [5] to include network coding to facilitate

scheduling. However, in [5] the authors do not explicitly

consider multiple flows, fairness, or scheduling, and in fact

show that the performance benefit drops as number of flows

increases.

One of the works most similar to our own is the opti-

mization framework for opportunistic routing that minimize

power consumption, presented in [19], which shows significant

benefits over [4]. Nevertheless, [19] considers neither network

coding, nor the TCP-like primal-dual rate adaptation. It is not

clear how to generalize [19] to use network coding with finite

generation size and to derive a generation scheduling policy

analog to (34).

Another related paper is the work on energy-efficient oppor-

tunistic network coding [30], which use a similar formulation

3Note that an additional performance drop for finite generation size mayoccur due to imperfect signaling, but we do not consider it in our model.

of an optimization problem, which can further be extended to

intersession network coding. [30] does not consider a problem

of a limited generation size, TCP dynamics, nor a suboptimal

rate allocation algorithms and has a less general interference

model. Cross-layer design for network coding with unicast or

multi-cast sessions is considered in [9]; however, [9] considers

only stability and not any form of rate maximization. End-to-

end rate maximization of a single flow is also considered in

[31].

Several theoretical analysis of linear network coding algo-

rithms for unicast sessions have been performed [16], [10].

Network coding for unicast sessions is used also in COPE

[32]. Compared to COPE, we perform encoding operations

only between packets of the same flow; in that respect our

approach is orthogonal to COPE. Optimal control of inter-

session network coding is presented for example in [33];however, it does not consider opportunistic routing and its

inherent diversity.

Our work is an example of cross-layer optimization, and we

have built on top of and extended exiting research. Cross-layer

design in wireless is a widely research topics (see [7], [13],

[14] and references therein). Optimizing network performance

in terms of network utility is originally proposed in [22];

see [11], [14] for an overview. Our primal-dual approach is

similar to [11], where it is shown that it can capture different

versions of TCP. None of the algorithms in [7], [13], [14], [11]

consider opportunistic routing, broadcast diversity and intra-

session network coding.

VII. CONCLUSIONS

This paper proposes an optimization framework for addressing

questions of multi-path routing in wireless mesh networks. We

have extended previous work by incorporating the broadcast

nature of wireless and simultaneously addressing fairness

issues. Implicit in our approach is the use of network coding,

which enables us to define notions of credits that are associated

with number of packets in a generation, rather than specific

packets. Using our framework we show that our algorithm

significantly outperforms single-path routing and MORE [5].


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12

When scheduling is pre-determined by a MAC protocol,

such as by random scheduling or 802.11-like scheduling,

we have shown how our approach leads to a distributed

heuristic, which still outperforms existing approaches. Using

a simulation results on a realistic topology, we found in our

examples that for 802.11b, using the maximal rate is optimal,

but for 802.11a this was not the case. We have addressed

some of the practical issues associated with having a finite

generation size for network codes.

Our primal-dual rate adaptation can be used to model

window-based flow control schemes, such as TCP. The per-

formance of applications that run on top of our system and

use TCP is an interesting open problem. Another interesting

direction is to analyze the performance of our protocol with

more realistic signaling schemes.

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mesh networks for all-wireless offices, in ACM/Usenix MobiSys, 2006.[3] M. Caesar, M. Castro, E. Nightingale, G. OShea, and A. Rowstron,Virtual ring routing: Network routing inspired by DHTs, in ACMSigComm, 2006.

[4] S. Biswas and R. Morris, ExOR: opportunistic multi-hop routing forwireless networks, in ACM SIGCOMM, 2005.

[5] S. Chachulski, M. Jennings, S. Katti, and D. Katabi, Trading structurefor randomness in wireless opportunistic routing, in ACM SigComm,2007.

[6] B. Radunovic, C. Gkantsidis, P. Key, S. Gheorgiu, W. Hu, and P. Ro-driguez, Multipath code casting for wireless mesh networks, in MSR-TR-2007-68, March 2007.

[7] L. Georgiadis, M. J. Neely, and L. Tassiulas, Resource allocation andcross-layer control in wireless networks, Foundations and Trends in

Networking, vol. 1, no. 1, pp. 1144, 2006.

[8] R. Ahlswede, N. Cai, S. R. Li, and R. W. Yeung, Network informationflow, IEEE Transactions on Information Theory, 2000.

[9] T. Ho and H. Viswanathan, Dynamic algorithms for multicast withintra-session network coding, in 43rd Allerton Annual Conference onCommunication, Control, and Computing, 2005.

[10] D. Lun, M. Medard, and R. Koetter, Network coding for efficient wire-less unicast, in IEEE International Zurich Seminar on Communications,February 2006.

[11] R. Srikant, The Mathematics of Internet Congestion Control.Birkhauser, 2003.

[12] A. Eryilmaz and R. Srikant, Joint congestion control, routing and macfor stability and fairness in wireless networks, IEEE Journal on Selected

Areas in Communications, vol. 24, no. 8, pp. 15141524, August 2006.

[13] X. Lin, N. B. Shroff, and R. Srikant, A tutorial on cross-layer opti-mization in wireless networks, IEEE J. on Selected Areas in Comm.,vol. 24, no. 8, Aug 2006.

[14] M. Chen, S. Low, M. Chiang, and J. Doyle, Cross-layer congestioncontrol, routing and scheduling design in ad hoc wireless networks, in

INFOCOM, 2006.

[15] G. Sharma, R. Mazumdar, and N. Shroff, On the complexity ofscheduling in wireless networks, in Proc. MOBICOM, 2006.

[16] D. Lun, M. Medard, and M. Effros, On coding for reliable communi-cation over packet networks, in Proc. 42nd Allerton Conference, 2004.

[17] L. Tassiulas and A. Ephremides, Stability properties of constrainedqueueing systems and scheduling policies for maximum throughput inmultihop radio networks, IEEE Trans. on Automatic Control, vol. 37,no. 12, 1992.

[18] M. Neely, E. Modiano, and C. Rohrs, Dynamic power allocation androuting for time-varying wireless networks, IEEE Journal on Selected

Areas in Communications, vol. 23, no. 1, pp. 89103, January 2005.[19] M. Neely, Optimal backpressure routing for wireless networks with

multi-receiver diversity, in Conference on Information Science andSystems, March 2006.

[20] D. Bertsekas and J. Tsitsiklis, Parallel and Distributed Computation:Numerical Methods. Prentice Hall, 1989.

[21] B. Radunovic, C. Gkantsidis, G. Gunawardena, and P. Key, Horizon:Balancing tcp over multiple paths in wireless mesh network, in Proc.

MOBICOM, 2008.

[22] F. P. Kelly, A. Maulloo, and D. Tan, Rate control in communicationnetworks: shadow prices, proportional fairness and stability, Journal ofthe Operational Research Society, vol. 49, pp. 237252, 1998.

[23] S. Boyd and L. Vandenberghe, Convex Optimization. Cambri dgeUniversity Press, 2004.

[24] S. Chachulski, M. Jennings, S. Katti, and D. Katabi, MORE: A networkcoding approach to opportunistic routing, in MIT-CSAIL-TR-2006-049 ,2006.

[25] P. Chaporkar, K. Kar, and S. Sarkar, Throughput guarantees throughmaximal scheduling in wireless networks, in Proc. Allerton, 2005.

[26] E. Modiano, D. Shah, and G. Zussman, Maximizing throughput inwireless networks via gossiping, in Proc. ACM SIGMETRICS / IFIPPerformance06, June 2006.

[27] A. K. Miu, H. Balakrishnan, and C. E. Koksal, Improving LossResilience with Multi-Radio Diversity in Wireless Networks, in 11th

ACM MOBICOM Conference, Cologne, Germany, September 2005.

[28] R. Gummadi, R. Patra, H. Balakrishnan, and E. Brewer, InterferenceAvoidance and Control, in 7th ACM Workshop on Hot Topics in

Networks (Hotnets-VII), Calgary, Canada, October 2008.

[29] L. Huang, B. Johnson, D. Tadas, and M. Stoler, 802.11a performanceover various channels, in IEEE 802.11-03/0682-00-000k, 2003.

[30] T. Cui, L. Chen, and T. Ho, Effcient opportunistic network coding forwireless networks, in Proc. INFOCOM, 2008.

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[32] S. Katti, H. Rahul, W. Hu, D. Katabi, M. Medard, and J. Crowcroft,XORs in the air: Practical wireless network coding, in ACM SigComm,2006.

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APPENDIX

Before proving Theorem 2 we first introduce the following

lemmaLemma 1: The following equalities and inequalities hold

for any t:

i,jN

cC

ycij (ci

cj ) =

cC

fc ci , (37)

ciJxciJ =

ciJ

jJ

ycij , (38)

iN,JP(N)

cC

ciJxciJ(t)

iN,JP(N)

cC

ciJxciJ,

(39)

iN,JP(N)

cC

c

iJx

c

iJ(t)

i,jN,cC

c

ij y

c

ij , (40)i,jN,cC

ycij (qci (t) q

cj(t)) =

cC

fc qcSrc(c)(t), (41)

iN,JP(N)

wciJ(t)jJ

ycij(t) =i,jN

wcij(t)ycij(t), (42)

iN,JP(N)

cC

wciJ(t)xciJ(t)

i,jN,cC

wcij(t)ycij (43)

Proof: Equalities (37) and (38) follow directly from (6)

and (7). From (4) we further have

cC x

ciJ(t) = CiJ(t);


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13

Together with (10) this implies

i,J,c

ciJxciJ(t)

i,J,c

J

ciJCiJ(t)

i,J,c

J

ciJCiJ

=i,J,c

ciJxciJ

which proves (39). From (38) and (39) we derive (40). Since

by definition qcDst(c)(t) = 0 we havei,jN,cC

ycij (t)(qci (t) q

cj(t)) =

cC,jN

ycSrc(c)jqcSrc(c)(t)

from which we derive (41). Equality (42) follows from the

definition of wcij . Also, by definition of scheduling (13)-(18),we have

i,J,c

wciJ(t)xciJ(t) i

maxc

J

wciJ(t)CiJ(CiJ) (44)

i,J,c

wciJ(t)xciJ (45)

which yields (43).

A. Proof of Theorem 2

Proof: We will follow the idea of the proof of Theorem

2 from [12]. First, let us define Lyapunov function

W(f,q,w) =1

2

cC

(fc fc )

2 +1

2

iN,cC

(qci ci )

2

+1

2

iN,JP(N)

cC

(wciJ ciJ)

2. (46)

We want to show that the derivative W 0. For brevity,we define fci (t) = fc(t)1{i=Src(c)}. Using (20), (21) and (22)gives derivative of W as

W(f(t),q(t),w(t))

= c

(fc(t) fc )(U

c(fc(t)) q

c

Src(c)

(t))fc(t)0

+i,c

(qci (t) ci )

fci (t) +

j

ycji(t) k

ycik(t)

qci (t)0

+i,J,c

(wciJ(t) ciJ)

jJ

ycij(t) xciJ(t)

wciJ(t)0

. (47)

As in (10)-(13) from [12] we have that, when qci (t) < 0 thederivative qci (t) is by definition positive; also

ci 0. The

same holds for wciJ(t) and fc(t) and we can upperbound the

derivative

W(f(t),q(t),w(t)) c

(fc(t) fc )(U

c(fc(t)) q

cSrc(c)(t)

+i,c

(qci (t) ci )

fci (t) +

j

ycji(t) k

ycik(t)

+i,J,c

(wciJ(t) ciJ)jJ

ycij(t) xciJ(t)

(48)Let us further add and subtract Uc(f

c ) =

cSrc(c), as in [12],

to obtain

W(f(t),q(t),w(t)) c

(fc(t) fc )(U

c(fc(t)) U

c(f

c ))

+i,c

ci

k

ycik(t) j

ycji(t) fci

+i,c

qci

(t)j

ycji

(t) k

ycik

(t) + fci

+i,J,c

(wciJ(t) ciJ)

jJ

ycij(t) xciJ(t)

.

Due to concavity of Uc we have (fc(t) fc )(Uc(fc(t))

Uc(fc )) 0. Next, let us pick any set of link rates {y

cij }i,j,c

that correspond to the optimal flow allocation {fc }c. Weexpand fc using (37) and (41) to obtain

W(f(t),q(t),w(t)) i,j,c

(ci cj )(y

cij(t) y

cij )

+i,j,c

(qci q

cj )(y

cij y

cij(t))

+i,J,c

(wciJ(t) ciJ)

jJ

ycij(t) xciJ(t)

.

Let us denote zcij = ci

cj

cij . Then, from (40), (42)

and (43) we have

W(f(t),q(t),w(t))

i,j,c

(ycij(t) ycij )

(ci cj cij ) (qci (t) qcj(t) wcij(t)) (49)=i,j,c

(ycij(t) ycij )(z

cij z

cij(t)) (50)

(a)=i,j,c

ycij(t)zcij (y

cij(t) y

cij )z

cij(t) (51)

(b)

i,j,c

(ycij(t) ycij )z

cij(t)

(c)

0. (52)

where (a) follows from KKT and the fact that ycij zcij = 0,

(b) from (9) and (c) from the fact that whenever qci (t) > 0and zcij(t) > 0 then y

cij(t) can be made arbitrarily large in the

fluid model limit as the slot length goes to zero.


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14

Hence we have that W(f(t),q(t),w(t)) 0 for all f(t) >0,q(t) > 0,w(t) 0. Let us define

Q(t) =

(q,w) |

i,j,c

(ycij(t) ycij )(z

cij z

cij(t))

(53)

Let us define E = {(f,q,w) | W(f,q,w) = 0}. It is easy tosee from (50) that E Q. We can further apply LaSalles

invariance principle as in [12] to show that f(t) converges tof and (q(t),w(t)) converges to Q(t).

However, set Q(t) is not bounded in general. If link (i, j) isactive for flow c then for every t, ycij(t) > 0 we have q

cj(t) +

wcij(t) = qci (t)z

cij . If the maximum node degree in a network

is D, we have that qcj(t) + wcij(t) q

ci (t) z

cij + 2DT. Since

qcSrc(c) converges to Uc(f

c ) we see that queues q

ci (t), q

cj(t)

and wcij(t) are bounded for all active links (i, j) of each flowc.

B. Derivation of (33)-(35)

Let us write modified queue evolution equations:

qci (t, g) =fci (t, g) +

j

ycji(t, g) j

ycij(t, g)qci (t,g)0

wciJ(t, g) =

jJ

ycij(t, g) xciJ(t, g)

wciJ(t,g)0

.

Note that (20) and (21) do not hold anymore. Consequently,

we cannot claim that (48) follows from (47) and the proof of

Theorem 2 cannot be applied.

We first consider qci (t, g). We see from (33) that ycik(t, g) >

0 only ifqci (t, g) > 0. Thus, the exact queue evolution is givenby

qci (t, g) = f

ci (t, g) +

j

ycji(t)

k

ycik(t).

Next, let us look at the evolution of wciJ(t, g). We have thatwciJ(t, g) = 0 only if w

ciJ(t, g) = 0 and x

ciJ(t, g) > 0, thus if

g = g(c, t). Therefore, we can write

wciJ(t, g) jJ

ycij(t) xciJ(t, g) 1{wciJ(t,g)0},

wciJ(t) jJ

ycij(t) xciJ(t) 1{wciJ(t,g(c,t))0}.

and from (47) we can write

W(f(t),q(t),w(t))

c

(fc(t) fc )(Uc(fc(t)) qcSrc(c)(t)) (54)

+i,c

(qci (t) ci )

fci (t) +

j

ycji(t) k

ycik(t)

(55)

+i,J,c

(wciJ(t) ciJ)

jJ

ycij(t) xciJ(t)

(56)

+i,J,c

wciJ(t)xciJ(t) 1{wciJ(t,g(c,t))

41. toward practical opportunistic routing with intra-session network coding for mesh networks

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